PHYSICAL REVIEW B 76, 085128 !2007"
Boundary Green’s function for spin-incoherent interacting electrons in one dimension
1Department
Paata Kakashvili1,2 and Henrik Johannesson3
of Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
of Physics and Astronomy, Rice University, 6100 Main Street, Houston, Texas 77005, USA
3
Department of Physics, Göteborg University, SE-412 96 Göteborg, Sweden
!Received 2 May 2006; revised manuscript received 12 June 2007; published 31 August 2007"
2Department
The spin-incoherent regime of one-dimensional electrons has recently been explored using the Bethe ansatz
and a bosonized path integral approach, revealing that the spin incoherence dramatically influences the correlations of charge excitations. We here introduce a bosonization scheme for strongly interacting electrons,
allowing us to generalize the description to account for the presence of an open boundary. By calculating the
single-electron Green’s function we find that the charge sector power-law scaling is highly sensitive to the
boundary. Our result allows for a detailed description of the crossover between boundary and bulk regimes. We
predict that scanning tunneling microscopy on a spin-incoherent system will pick up oscillations in the differential tunneling conductance as a function of the applied voltage V at “intermediate” distances x from a real or
a dynamically generated boundary. The wavelength of the oscillations, !vc / x, probes the speed vc of the
charge excitations, and therefore the strength of the electron-electron interaction.
DOI: 10.1103/PhysRevB.76.085128
PACS number!s": 71.10.Pm, 71.27."a, 73.21.#b
I. INTRODUCTION
The spin-incoherent regime of one-dimensional strongly
interacting, very low-density electrons has recently attracted
a lot of interest.1 For zero temperature the kinetic energy
of an electron can be estimated by the Fermi energy
EF = !!$n"2 / 8m. For low densities, n % aB−1, this energy is
small compared to the Coulomb potential energy e2n / & !with
& being the dielectric constant and aB = &$2 / me2 the effective
Bohr radius of the material". In the limit of low densities the
system turns into a Wigner crystal,2 which—in a classical
picture—can be viewed as a system of electrons placed equidistantly so as to minimize the potential energy. Quantum
fluctuations induce an exponentially small antiferromagnetic
spin exchange J ' 0 between the electrons. In the case when
J is the smallest energy scale, J % T % EF, the spin exchange
can no longer support collective spin excitations, and the
system is driven to a spin-incoherent regime. The physics of
the spin-incoherent regime has been addressed using the Bethe ansatz3,4 and a bosonized path integral approach.5 Surprisingly, it was found that the spin incoherence dramatically
influences the correlations of charge excitations, leading to a
power-law decay in the charge sector with an interactiondependent nonunitary exponent.
In this paper we generalize the description in Ref. 5 to
account for the presence of an open boundary. For this purpose we introduce a bosonization scheme, valid for strongly
interacting electrons, and presented in Sec. II. In Sec. III we
then derive the single-electron Green’s function in the presence of an open boundary condition. Scanning tunneling microscopy !STM" as a possible experimental probe of the
spin-incoherent regime is discussed in Sec. IV. The last section contains a brief summary of our results.
II. BOSONIZATION IN THE STRONG COUPLING
REGIME
H=
# $
dx
%
1
p2
+ mns2!!xu"2 .
2mn 2
!1"
Here u!x" is the displacement of the medium and p!x" is the
momentum density. Density fluctuations are given by
(n = −n!xu, and s plays the role of the speed of density
waves.
The classical Hamiltonian in Eq. !1" can be straightforwardly quantized by imposing a canonical commutation relation between u!x" and p!x",
&u!x",p!x!"' = i(!x − x!".
!2"
The resulting quantum Hamiltonian describes the propagation of density fluctuation in the Wigner crystal, and can
be written in terms of a bosonic field )c!x" and its conjugate
momentum *c!x", connected to u!x" and p!x" by
u!x" = −
We here introduce a bosonization scheme for strongly interacting electrons, for which the applicability of an ordinary
1098-0121/2007/76!8"/085128!5"
bosonization6 becomes questionable. In the strong-coupling
regime, the electrons are localized at the lattice sites of the
Wigner crystal, and the usual procedure of linearizing the
spectrum around the Fermi points ±kF and then expanding
the electron fields in left and right movers is no longer justified. In what follows we show that one can nonetheless
perform an “effective” bosonization, valid for low energies
and large distances.
To begin with, it can be shown that a bosonic Hamiltonian
correctly describes the low-energy properties of a onedimensional !1D" Wigner crystal.7 To justify this statement
we turn to the classical picture and describe a 1D Wigner
crystal as a system of electrons vibrating around their equilibrium positions. This is very similar to the classical description of phonons as lattice vibrations. For low energies
!long-wavelength limit" the vibrations can be described by
elasticity theory, with the energy of the system expressed by
the Hamiltonian
(2
(!n )c!x",
p!x" = −
(!n
(2 *c!x".
!3"
It follows that
085128-1
©2007 The American Physical Society
PHYSICAL REVIEW B 76, 085128 !2007"
PAATA KAKASHVILI AND HENRIK JOHANNESSON
H=
vc
2
# $
dx Kc*2c +
%
1
! ! x) c" 2 ,
Kc
!4"
vF
,
s
!5"
where
vc = s,
Kc =
with vc the speed of the propagating charge density fluctuations, and with vF = !n / 2m the Fermi velocity for noninteracting spinful electrons.
We have thus managed to write the Hamiltonian for a 1D
Wigner crystal in bosonized form although we did not know
the bosonization formula which connects electron and boson
fields. We next derive such a formula, applying a kind of
“reverse engineering” to the bosonic description of the
Wigner crystal above. First we observe that in the J → 0
limit, electrons behave like spinless fermionic particles, i.e.,
only charge degrees of freedom survive and spin degrees of
freedom simply define the huge degeneracy of the ground
state. We may thus omit the spin index and write
+!x" = ,−!x" + ,+!x",
,!!x" )
(2!. e
˜ x !i(/- !x"
!ik
F
c!
e
,
!= ±,
!7"
and
H=
1
(2!. e
vc
2
˜ x !i(4!- !x"
!ik
!
F
# $
e
dx K*2 +
,
!=±
%
1
! ! x) " 2 ,
K
0 = +†!x"+!x" = n +
(
2
! x) c ,
!
!8"
with the product of electron fields defined by point splitting,6
and where n is the average electron density and !(2 / !"!x)c
is a fluctuation term &cf. Eqs. !1" and !3"', we find that
/ = 8! .
!9"
We have here used that the fast oscillating nonchiral terms
,±†,1 contained in +†+ effectively average out to zero over
large distances, and hence can be neglected in the longwavelength limit. Also note that the doubling of the Fermi
momentum in Eq. !9" is in agreement with our description
of spinless fermions, since now only a single fermion can
occupy a given momentum state. As can be easily verified,
the case of noninteracting spinless fermions corresponds to
Kc = 1 / 2. It follows that in this bosonization scheme interactions are absorbed in two steps: Local interactions among the
spinful electrons are incorporated in a free Hamiltonian for
spinless fermions !with Kc = 1 / 2", while long-range interactions renormalize the value of Kc !away from Kc = 1 / 2". To
see this explicitly, note that for a delta function interaction
among the electrons—corresponding to the low-energy,
long-wavelength limit of the infinite-U 1D Hubbard
!10"
!11"
respectively, with K * 2Kc = 2vF / s = ṽF / s !in agreement with
the doubling of the Fermi momentum k̃F".
III. BOUNDARY GREEN’S FUNCTION
The bosonization procedure presented in the previous section can easily be adapted to the case when a boundary is
present. Imposing an open boundary condition !OBC" at the
end, x = 0, of a semi-infinite system with x 2 0, we have that
+!0" = ,−!0" + ,+!0" = 0.
!12"
Analytically continuing the chiral fermion operators to negative coordinates,
,+!x" = − ,−!− x",
with . a short-distance cutoff, and where k̃F and / are to be
determined. Using that the density operator is given by
k̃F = !n,
,!!x" )
!6"
where ,−!x" &,+!x"' is the part of the electron operator
which contains negative &positive' momenta. In analogy
with ordinary bosonization of spinless fermions6 we introduce two chiral boson fields -c− and -c+, connected to )c by
)c = -c− + -c+. We can then write an effective bosonization
formula, valid for low energies,
1
model2—the description as noninteracting spinless fermions
becomes exact !with Kc = 1 / 2".
To obtain a more conventional parametrization !where a
unit value of an effective “charge parameter” K corresponds
to the case of noninteracting spinless fermions" we define a
bosonic field ) * (2)c with conjugate momentum *
* *c / (2. Then the bosonization formula in Eq. !7" and the
Hamiltonian in Eq. !4" take the forms
!13"
and then following the standard procedure for an
OBC8—using our Eq. !10"—we obtain a bosonization formula for strongly interacting spin-incoherent electrons with
an OBC,
,−!x" )
1
(2!. e
−ik̃Fx −i(4!&cosh!3"-̃−!x"−sinh!3"-̃−!−x"'
e
. !14"
Here e23 = K, with -̃− = cosh!3"-− − sinh!3"-+ governed by
the free chiral boson Hamiltonian
H=
vc
2
#
dx&!x-̃−!x"'2 .
!15"
To calculate the zero-temperature single-electron Green’s
function in the presence of the OBC we follow the path
integral approach introduced in Ref. 5. The averaging over
spin introduces a factor 2−N!x,x!,4" in the expression for the
Green’s function, where N!x , x! , 4" samples the number of
electrons, or equivalently, the number of noncrossing world
lines in the interval x − x!. In the spin-incoherent regime, here
realized by first taking J → 0 and then T → 0, the spin configurations all carry the same weight, and the probability that
the world lines in the interval x − x! all have the same spin !as
required for a nonzero contribution to the low-energy
Green’s function" becomes equal to 2−N!x,x!,4". Given this, the
calculation of the Green’s function G!x , x! , 4" translates into
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PHYSICAL REVIEW B 76, 085128 !2007"
BOUNDARY GREEN’S FUNCTION FOR SPIN-INCOHERENT…
the calculation of four correlators for spinless time-boosted
fermions, with the spin averaging factor 2−N!x,x!,4" properly
inserted,9
G!x,x!, 4" = +2−N!x,x!,4"+!x, 4"+†!x!,0",
= +2−N!x,x!,4",−!x, 4",−† !x!,0",
+ +2−N!x,x!,4",+!x, 4",+†!x!,0",
+ +2−N!x,x!,4",−!x, 4",+†!x!,0",
+ +2−N!x,x!,4",+!x, 4",−† !x!,0",.
!16"
The operator N!x , x! , 4" can be expressed as
N!x,x!, 4" = n-x − x!- +
1
(! &)!x, 4" − )!x!,0"',
!17"
where n is the average electron !or world line" density, and
with !1 / (!"&)!x , 4" − )!x! , 0"' = &cosh!3" + sinh!3"'&-̃−!x , 4"
− -̃−!−x , 4" − -̃−!x! , 0" + -̃−!−x! , 0" / (!' the fluctuation term.
The correlators in Eq. !16" are straightforwardly calculated by first using Eq. !13" to replace all occurrences of
right-moving fermion fields by left movers, and then applying the bosonization formula !14". Introducing relative and
center-of-mass coordinates, r = x − x! and R = !x + x!" / 2, respectively, we obtain for the G−− piece of the Green’s
function
G− −!x,x!, 4" * +2−N!x,x!,4",−!x, 4",−† !x!,0",
=
1 −!ln 2/!"k̃ -r- −ik̃ r i5
1
1
F e −−
F e
e
2!.
!. sgn 4 + vc4 + ir" &!. sgn 4 + vc4"2 + r2'−261+262
7
.(
where
5−− =
&.2 + !2R + r"2'&.2 + !2R − r"2'
!. sgn 4 + vc4"2 + 4R2
/
261+263
!18"
,
0
0
ln 2
&.2 + !2R + r"2'&.2 + !2R − r"2'
!. sgn 4 + vc4"2 + r2
K ln
+ 2 ln
2
2 2
4!
&!. sgn 4 + vc4" + 4R '
.2
+
1
1
ln 2
&. − i!r + 2R"'&. − i!r − 2R"'
. sgn 4 + vc4 + ir
ln
,
+ 2 ln
4!
&. + i!r + 2R"'&. + i!r − 2R"'
. sgn 4 + vc4 − ir
!19"
and with the exponents given by 61 = K8 ! ln!2 "2, 62 = 81 ! K1 + K − 2", 63 = 81 ! K1 − K".
The G− + part of the Green’s function reads
G− +!x,x!, 4" * +2−N!x,x!,4",−!x, 4",+†!x!,0",
=
1
1 −!ln 2/!"k̃ -r- −i2k̃ R i5
1
F e
F e −+
e
2!.
!. sgn 4 + vc4 + 2iR" &!. sgn 4 + vc4"2 + 4R2'261+262
72(&.2 + !2R + r"2'&.2 + !2R − r"2'&!. sgn 4 + vc4"2 + r2'3261
0
(&.2 + !2R + r"2'&.2 + !2R − r"2'
!. sgn 4 + vc4"2 + r2
1
263
,
!20"
function are immediately obtained from Eqs. !18"–!20" by
using that
where
5−+ =
ln 2
.2 + !2R + r"2
K ln 2
4!
. + !2R − r"2
+
0
G+ +!x,x!, 4" = G−* −!x,x!, 4",
ln 2
&. − i!r + 2R"'&. − i!r − 2R"'
ln
4!
&. + i!r + 2R"'&. + i!r − 2R"'
+ 2 ln
1
. sgn 4 + vc4 + ir
.
. sgn 4 + vc4 − ir
G+ −!x,x!, 4" = G−* +!x,x!, 4".
!21"
The expressions for the G+ + and G+ − pieces of the Green’s
!22"
Having derived the full expression for the boundary
Green’s function we can study boundary and bulk regimes by
taking the proper limits. The bulk regime is defined by R
8 vc4 , r. For the chiral G− − piece we obtain
085128-3
PHYSICAL REVIEW B 76, 085128 !2007"
PAATA KAKASHVILI AND HENRIK JOHANNESSON
G− −!x,x!, 4" = +2−N!x,x!4",−!x, 4",−† !x!,0",
1.4
1 −!ln 2/!"k̃ -r- −ik̃ r i5
1
F e
F e −−
e
2!.
!. sgn 4 + vc4 + ir"
7
1
,
&!. sgn 4 + vc4"2 + r2'−261+262
1.2
ρuni [arb.unit]
=
1.6
!23"
in agreement with the result for the bulk Green’s function
derived in Refs. 3 and 5. In contrast to the chiral part G− −
!G+ +" of the Green’s function, G− + !G+ −" decays with a
power law and vanishes in the bulk limit, as it must
1
G− +!x,x!, 4" 4 K .
R
G− −!x,x!, 4" 4 G− +!x,x!, 4" 4
1
.
41/K
!25"
This agrees with the result in Ref. 10 !see also Ref. 11".
Importantly, our general result allows for a study of the
crossover between boundary and bulk regimes.
IV. STM RESPONSE IN THE SPIN-INCOHERENT REGIME
In this section we discuss a possible experimental probe
of the spin-incoherent regime, using scanning tunneling microscopy !STM". Assuming that the STM tip couples only to
the conduction electrons, the differential tunneling conductance dI!V , x" / dV is directly proportional to the local electron tunneling density of states !LDOS" 0!V , x", where V is
the applied voltage, and x is the position of the tip as measured from the boundary.12 The boundary may here be one of
the end points of a quantum wire or a metallic nanotube, or
generated dynamically13 by an impurity in the system !an
antidot, or a deep “core level” hole created by an x-ray
photon10". The LDOS is related to the boundary Green’s
function in Eq. !16" by
0!x, 9" ) −
1
Im GR!x, 9",
!
!26"
where the retarded Green’s function GR!x , 9" is obtained
from the Fourier transform G!x , 9n" of G!x , x , 4" in Eq. !16"
by analytically continuing i9n → 9 + i:+. The " sign in Eq.
!26" is a reminder that the expression for the Green’s function in Eq. !16" as derived above becomes exact only in the
long-wavelength limit &cf. the text after Eq. !9"'. The LDOS
has a uniform rapidly oscillating part originating from
the chiral !nonchiral" Friedel-type terms G− −/+ +!x , x , 4"
&G− +/+ −!x , x , 4"' in G!x , x , 4",
0!x, 9" = 0uni!x, 9" + cos!2k̃Fx"0osc!x, 9".
!27"
Figure 1 shows 0uni for three choices of distance from the
boundary, obtained from Eq. !16" by numerically computing
the integrals which define the analytically continued Fourier
0.8
0.6
k̃F x = π
0.4
k̃F x = 5π
0.2
!24"
For the boundary case, defined by vc4 8 r , R, the chiral
and nonchiral parts of the Green’s function show the same
behavior, with the asymptotic scaling
1
0
0
k̃F x = 10π
0.2
0.4
0.6
ω/ k̃F vF
0.8
1
1.2
FIG. 1. Uniform part of the LDOS, 0uni, as a function of energy
9 for different boundary-to-tip distances x. Oscillations for intermediate distances are a result of interference of incoming and reflected
charge modes at the boundary.
transforms. The most interesting feature is the oscillation
pattern seen for “intermediate” distances where 9x 4 vc !i.e.,
the crossover regime between boundary and bulk behavior".
This pattern is due to the interference of incoming and reflected charge modes at the boundary. The oscillations, of
wavelength !vc / x, give immediate information about the
speed vc of the charge modes, and therefore about the
strength of the electron-electron interaction. This unique signal of the spin-incoherent phase should be readily obtained
in STM experiments as an oscillation pattern in the differential tunneling conductance as a function of the applied voltage V !which fixes the frequency for which the LDOS is
probed". Let us here point out that the oscillations are present
at any distance from the boundary, but that in the boundary
&bulk' regime, defined by 9x % vc &9x 8 vc', the wavelength
may become too long &short' to be possible to resolve experimentally. !Cf. Fig. 1 where the wavelength for the oscillation for the case k̃Fx = ! is roughly twice larger than the
displayed frequency interval." It is important to stress that
the Friedel terms &the last two terms in Eq. !16"' do not
change the result qualitatively. This is in contrast to an ordinary Luttinger liquid where the interference of charge and
spin modes with different speeds produce an oscillation pattern with very different properties.14 In the case of a spinincoherent system there is only one propagating charge mode
with a single speed vc, and the oscillation pattern of 0osc will
be the same as that of 0uni.
Let us conclude this section by mentioning that other experimental probes of spin incoherence have been suggested
in the literature, including momentum resolved tunneling experiments on cleaved-edge overgrowth quantum wires,15,16
and quantum interference experiments on specially designed
devices.11 It would be very interesting to explore how the
characteristic crossover behavior of the single-electron
Green’s function identified above may influence these proposed experiments.
V. CONCLUSIONS
We have calculated the exact single-electron boundary
Green’s function for one-dimensional spin-incoherent elec-
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PHYSICAL REVIEW B 76, 085128 !2007"
BOUNDARY GREEN’S FUNCTION FOR SPIN-INCOHERENT…
trons in the low-energy, long-wavelength limit, using a
bosonization scheme valid in the strong-coupling regime.
The Green’s function thus obtained correctly reproduces
known results in the bulk and extreme boundary regimes. As
revealed by the expressions in Eqs. !18" and !21", the charge
sector power-law scaling of the Green’s function is highly
sensitive to the presence of the boundary also at intermediate
distances away from it, where the center-of-mass coordinate
R 4 O!vc4". This feature will strongly influence the local tunneling density of states, and may be probed experimentally
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a review, see G. A. Fiete, Rev. Mod. Phys. 79, 801 !2007"
and references therein.
2
H. J. Schulz, G. Cuniberti, and P. Pieri, in Field Theories for
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9
In contrast to Ref. 5 we do not include the permutation factor
!−1"N in the expression for the Green’s function since it is au-
by scanning tunneling microscopy of one-dimensional conductors of spin-incoherent electrons.
ACKNOWLEDGMENTS
It is a pleasure to thank M. Zvonarev for an inspiring
discussion about spin-incoherent electrons, and G. Fiete for
helpful communications. This paper was supported by a
grant from the Swedish Research Council.
tomatically taken into account in our spinless fermion formulation. In Ref. 5, averages are carried out for spinless bosons,
requiring the permutation factor to be explicitly inserted &cf. the
discussion after Eq. 41 in Ref. 15'.
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!2005".
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085128-5